Problem 39 - Calculus for Biology and Medicine

Question

An urn contains three red and two blue balls. You remove two balls without replacement. What is the probability that the two balls are of a different color?

Step-by-Step Solution

Answer

The probability is \( \frac{3}{5} \).

Step 1: Understanding the Problem

We have an urn containing 3 red balls and 2 blue balls. We aim to find the probability that when we draw two balls without replacement, they are of different colors.

Step 2: Calculate Total Outcomes

First, determine the total number of ways to select 2 balls from the 5 balls in the urn. This can be calculated using combinations: \( \binom{5}{2} = 10 \).

Step 3: Calculate Favorable Outcomes for Different Colors

We need one red and one blue ball. First, select one red ball from 3 red balls: \( \binom{3}{1} = 3 \). Then, select one blue ball from 2 blue balls: \( \binom{2}{1} = 2 \). The total number of favorable outcomes = \( 3 \times 2 = 6 \).

Step 4: Calculate Probability

The probability that the two balls are of different colors is calculated by the ratio of favorable outcomes to total outcomes: \( \frac{6}{10} = \frac{3}{5} \).

Key Concepts

Combinations

Combinations are a powerful tool in probability and statistics used to determine the number of ways to choose items from a larger set when the order of selection does not matter. Imagine you have a set of 5 balls and you want to pick 2 of them. The formula for combinations is represented by \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. In our exercise, \( n = 5 \) (the total number of balls) and \( r = 2 \) (the number of balls we want to pick), so we use \( \binom{5}{2} \), which equals 10. This means there are 10 different ways to select 2 balls from 5, highlighting that combinations make it easy to manage scenarios where only the selection counts, not the order.

• Useful for calculating probabilities.
• Helps in scenarios where sequence is not important.

Probability theory

Probability theory is the mathematical framework for quantifying uncertainty. It's about predicting the likelihood of various outcomes. In the context of our urn with balls problem, probability is calculated by dividing the number of favorable outcomes by the total possible outcomes. We first need to know how many total outcomes are possible when picking 2 balls, which we determined to be 10 using combinations.The favorable outcomes are the number of ways to pick 2 balls of different colors, which we calculated as 6. Thus, the probability is found using the formula:\[\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{10} = \frac{3}{5}\]This tells us that there's a 60% chance of picking two balls of different colors.

• Helps quantify chances of outcomes.
• A core principle in making predictions and understanding random processes.

Combinatorics

Combinatorics is a broader field of mathematics focused on counting, arranging, and finding patterns. It includes combinations, permutations, and other counting techniques. Combinatorics provides the tools needed to solve problems involving the arrangement of objects. In the urn problem, combinatorics comes into play when determining the different ways we can pick the balls. We used combinations instead of permutations because the order in which we pick the balls doesn't matter. Given the different possible selections and their probabilities, combinatorics allows us to effectively count and understand the different possible scenarios.

• Essential for organizing and counting possibilities.
• Forms the backbone of probability and statistics problems.